Inscribed Rectangle under a Parabola

What is the area of the largest rectangle that you can inscribe between the parabola and the x-axis?
Using Geogebra to explore this problem.
  1. Use the left-window to enter the equation. It is important to use notation.
  2. Plot point A on the x-axis (if you did it right, it should be sky blue)
  3. Use the reflect tool to reflect A over the y-axis
  4. Use the left-window to plot the following two points. (x(A),f(x(A)) (x(A’),f(x(A’))
  5. Switch to the Arrow Tool. If this was done correctly, all 4 points should move when you drag point A.
  6. Use the Polygon tool to draw a polygon around the 4 points. Poly1 in the left-window shows the area of the rectangle.
  7. Drag A until the area is at a maximum (keep the rectangle above the x-axis)
  1. What is the maximum area?
  2. What value of x that maximized the area?
  3. What is the base length?
  4. What is the height?
  5. The area of a rectangle is . What is the base length in terms of x? What is the height in terms of x? What is the area?
  6. Find the derivative of the area and use this to find when the area of the rectangle has a maximum value.